Non-linear systems tend to be ugly, no matter what. But you might be able to simplify a bit by factoring:

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Plug "3" in for the "x - y", and simplify by dividing through by the 3. Then solve the resulting literal equation for, say, y in terms of x, using the Quadratic Formula:

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This gives you two solutions, presumably one from the "plus" and the other from the "minus" (which you can see on your calculator, if you graph Y1=X-3 and Y2=(X^3-387)^(1/3) in the same window). The solutions will be the places where a "half" above crosses the other line, y = x - 3, assuming there is a crossing. The solution to one "half" might start like this:

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This is a radical equation; you begin to solve by squaring both sides:

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Finish the simplification, and then verify that each x-value is "allowed" inside the square root in the equation with the "" above. Once you've found which, if any, of the values is allowable, back-solve (using "y = x - 3") for the corresponding y-values for that "half".

Then note that, due to the squaring, solving the other "half" should look very similar.